On the topological entropy of (a, b)-continued fraction transformations

We study the topological entropy of a two-parameter family of maps related to (a, b)-continued fraction algorithms and prove that it is constant on a square within the parameter space (two vertices of this square correspond to well-studied continued fraction algorithms). The proof uses conjugation to maps of constant slope. We also present experimental evidence that the topological entropy is flexible (i.e. takes any value in a range) on the whole parameter space.